Let us write #9x^2+4y^2-36x+24y+36=0# in general form. In general form equation is #(x-h)^2/a^2+(y-k)^2/b^2=1#, where #(h,k)# is center of ellipse and larger of #a# and #b# is half the major axis and smaller one is half the minor axis.
The equation can be written as
#9(x^2-4x)+4(y^2+6y)=-36#
or #9(x^2-4x+4)+4(y^2+6y+9)=-36+36+36#
or #9(x-2))^2+4(y+3)^2=36#
or #(x-2)^2/2^2+(y+3)^2/3^2=1#
Hence center of ellipse is #(2,-3)#, major axis is #6# parallel to #y#-axis and minor axis is #4# parallel to #x#-axis.
Hence, we get eccentricity #e=sqrt(1-4^2/6^2)=sqrt(1-4/9)=sqrt5/3#
and focii are at a distance of #be# i.e. #3xxsqrt5/3# i.e. #sqrt5# on either side of center parallel to major axis i.e.
focii are #(2,-3+sqrt5)# and #(2,-3-sqrt5)#.
graph{(9x^2+4y^2-36x+24y+36)((x-2)^2+(y+3)^2-0.03)((x-2)^2+(y+3+sqrt5)^2-0.03)((x-2)^2+(y+3-sqrt5)^2-0.03)=0 [-6, 10, -7, 1]}
